Finding Gapped Palindromes Online

Abstract

A string s is said to be a gapped palindrome iff $s = xyx^R$ for some strings x, y such that $|x| \ge 1$, $|y| \ge 2$, and $x^R$ denotes the reverse image of x. In this paper we consider two kinds of gapped palindromes, and present efficient online algorithms to compute these gapped palindromes occurring in a string. First, we show an online algorithm to find all maximal g-gapped palindromes with fixed gap length $g \ge 2$ in a string of length n in $O(n \log \sigma )$ time and O(n) space, where $\sigma$ is the alphabet size. Second, we show an online algorithm to find all maximal length-constrained gapped palindromes with arm length at least $A \ge 1$ and gap length in range $[ g _{\min }, g _{\max }]$ in $O(n(\frac{ g _{\max }- g _{\min }}{ A } + \log \sigma ))$ time and O(n) space. We also show that if $A$ is a constant, then there exists a string of length n which contains $\varOmega (n( g _{\max }- g _{\min }))$ maximal LCGPs, which implies we cannot hope for a significant speed-up in the worst case.